3.314 \(\int x^4 (A+B x) \sqrt{a+c x^2} \, dx\)

Optimal. Leaf size=152 \[ \frac{a^3 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{5/2}}+\frac{a^2 A x \sqrt{a+c x^2}}{16 c^2}+\frac{a \left (a+c x^2\right )^{3/2} (64 a B-105 A c x)}{840 c^3}+\frac{A x^3 \left (a+c x^2\right )^{3/2}}{6 c}-\frac{4 a B x^2 \left (a+c x^2\right )^{3/2}}{35 c^2}+\frac{B x^4 \left (a+c x^2\right )^{3/2}}{7 c} \]

[Out]

(a^2*A*x*Sqrt[a + c*x^2])/(16*c^2) - (4*a*B*x^2*(a + c*x^2)^(3/2))/(35*c^2) + (A
*x^3*(a + c*x^2)^(3/2))/(6*c) + (B*x^4*(a + c*x^2)^(3/2))/(7*c) + (a*(64*a*B - 1
05*A*c*x)*(a + c*x^2)^(3/2))/(840*c^3) + (a^3*A*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x
^2]])/(16*c^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.383644, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{a^3 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{5/2}}+\frac{a^2 A x \sqrt{a+c x^2}}{16 c^2}+\frac{a \left (a+c x^2\right )^{3/2} (64 a B-105 A c x)}{840 c^3}+\frac{A x^3 \left (a+c x^2\right )^{3/2}}{6 c}-\frac{4 a B x^2 \left (a+c x^2\right )^{3/2}}{35 c^2}+\frac{B x^4 \left (a+c x^2\right )^{3/2}}{7 c} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(A + B*x)*Sqrt[a + c*x^2],x]

[Out]

(a^2*A*x*Sqrt[a + c*x^2])/(16*c^2) - (4*a*B*x^2*(a + c*x^2)^(3/2))/(35*c^2) + (A
*x^3*(a + c*x^2)^(3/2))/(6*c) + (B*x^4*(a + c*x^2)^(3/2))/(7*c) + (a*(64*a*B - 1
05*A*c*x)*(a + c*x^2)^(3/2))/(840*c^3) + (a^3*A*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x
^2]])/(16*c^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 38.7896, size = 141, normalized size = 0.93 \[ \frac{A a^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{16 c^{\frac{5}{2}}} + \frac{A a^{2} x \sqrt{a + c x^{2}}}{16 c^{2}} + \frac{A x^{3} \left (a + c x^{2}\right )^{\frac{3}{2}}}{6 c} - \frac{4 B a x^{2} \left (a + c x^{2}\right )^{\frac{3}{2}}}{35 c^{2}} + \frac{B x^{4} \left (a + c x^{2}\right )^{\frac{3}{2}}}{7 c} + \frac{a \left (a + c x^{2}\right )^{\frac{3}{2}} \left (- 315 A c x + 192 B a\right )}{2520 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(B*x+A)*(c*x**2+a)**(1/2),x)

[Out]

A*a**3*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(16*c**(5/2)) + A*a**2*x*sqrt(a + c*x**
2)/(16*c**2) + A*x**3*(a + c*x**2)**(3/2)/(6*c) - 4*B*a*x**2*(a + c*x**2)**(3/2)
/(35*c**2) + B*x**4*(a + c*x**2)**(3/2)/(7*c) + a*(a + c*x**2)**(3/2)*(-315*A*c*
x + 192*B*a)/(2520*c**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.122527, size = 108, normalized size = 0.71 \[ \frac{105 a^3 A \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\sqrt{a+c x^2} \left (128 a^3 B-a^2 c x (105 A+64 B x)+2 a c^2 x^3 (35 A+24 B x)+40 c^3 x^5 (7 A+6 B x)\right )}{1680 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(A + B*x)*Sqrt[a + c*x^2],x]

[Out]

(Sqrt[a + c*x^2]*(128*a^3*B + 40*c^3*x^5*(7*A + 6*B*x) + 2*a*c^2*x^3*(35*A + 24*
B*x) - a^2*c*x*(105*A + 64*B*x)) + 105*a^3*A*Sqrt[c]*Log[c*x + Sqrt[c]*Sqrt[a +
c*x^2]])/(1680*c^3)

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 136, normalized size = 0.9 \[{\frac{A{x}^{3}}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{aAx}{8\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}Ax}{16\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{A{a}^{3}}{16}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{B{x}^{4}}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{4\,aB{x}^{2}}{35\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{8\,{a}^{2}B}{105\,{c}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(B*x+A)*(c*x^2+a)^(1/2),x)

[Out]

1/6*A*x^3*(c*x^2+a)^(3/2)/c-1/8*A*a/c^2*x*(c*x^2+a)^(3/2)+1/16*a^2*A*x*(c*x^2+a)
^(1/2)/c^2+1/16*A*a^3/c^(5/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+1/7*B*x^4*(c*x^2+a)^
(3/2)/c-4/35*a*B*x^2*(c*x^2+a)^(3/2)/c^2+8/105*B*a^2/c^3*(c*x^2+a)^(3/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + a)*(B*x + A)*x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.310207, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, A a^{3} c \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (240 \, B c^{3} x^{6} + 280 \, A c^{3} x^{5} + 48 \, B a c^{2} x^{4} + 70 \, A a c^{2} x^{3} - 64 \, B a^{2} c x^{2} - 105 \, A a^{2} c x + 128 \, B a^{3}\right )} \sqrt{c x^{2} + a} \sqrt{c}}{3360 \, c^{\frac{7}{2}}}, \frac{105 \, A a^{3} c \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (240 \, B c^{3} x^{6} + 280 \, A c^{3} x^{5} + 48 \, B a c^{2} x^{4} + 70 \, A a c^{2} x^{3} - 64 \, B a^{2} c x^{2} - 105 \, A a^{2} c x + 128 \, B a^{3}\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{1680 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + a)*(B*x + A)*x^4,x, algorithm="fricas")

[Out]

[1/3360*(105*A*a^3*c*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)) + 2*(24
0*B*c^3*x^6 + 280*A*c^3*x^5 + 48*B*a*c^2*x^4 + 70*A*a*c^2*x^3 - 64*B*a^2*c*x^2 -
105*A*a^2*c*x + 128*B*a^3)*sqrt(c*x^2 + a)*sqrt(c))/c^(7/2), 1/1680*(105*A*a^3*
c*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (240*B*c^3*x^6 + 280*A*c^3*x^5 + 48*B*a*c
^2*x^4 + 70*A*a*c^2*x^3 - 64*B*a^2*c*x^2 - 105*A*a^2*c*x + 128*B*a^3)*sqrt(c*x^2
+ a)*sqrt(-c))/(sqrt(-c)*c^3)]

_______________________________________________________________________________________

Sympy [A]  time = 21.6242, size = 216, normalized size = 1.42 \[ - \frac{A a^{\frac{5}{2}} x}{16 c^{2} \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{A a^{\frac{3}{2}} x^{3}}{48 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{5 A \sqrt{a} x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{A a^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 c^{\frac{5}{2}}} + \frac{A c x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + B \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x+A)*(c*x**2+a)**(1/2),x)

[Out]

-A*a**(5/2)*x/(16*c**2*sqrt(1 + c*x**2/a)) - A*a**(3/2)*x**3/(48*c*sqrt(1 + c*x*
*2/a)) + 5*A*sqrt(a)*x**5/(24*sqrt(1 + c*x**2/a)) + A*a**3*asinh(sqrt(c)*x/sqrt(
a))/(16*c**(5/2)) + A*c*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + B*Piecewise((8*a**
3*sqrt(a + c*x**2)/(105*c**3) - 4*a**2*x**2*sqrt(a + c*x**2)/(105*c**2) + a*x**4
*sqrt(a + c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c, 0)), (sqrt(a)*x**6/6,
True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.27558, size = 143, normalized size = 0.94 \[ -\frac{A a^{3}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, c^{\frac{5}{2}}} + \frac{1}{1680} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \,{\left (6 \, B x + 7 \, A\right )} x + \frac{6 \, B a}{c}\right )} x + \frac{35 \, A a}{c}\right )} x - \frac{32 \, B a^{2}}{c^{2}}\right )} x - \frac{105 \, A a^{2}}{c^{2}}\right )} x + \frac{128 \, B a^{3}}{c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + a)*(B*x + A)*x^4,x, algorithm="giac")

[Out]

-1/16*A*a^3*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2) + 1/1680*sqrt(c*x^2 +
a)*((2*((4*(5*(6*B*x + 7*A)*x + 6*B*a/c)*x + 35*A*a/c)*x - 32*B*a^2/c^2)*x - 105
*A*a^2/c^2)*x + 128*B*a^3/c^3)